Let $G$ be a finite abelian group. Is it true that the following element of the group ring ${\mathbb Z}[G]$: $$ \prod_{g\ne 1}(1g) $$ is nonzero?
Counterexample : $G=(\mathbb{Z}/2)^2$. The product is $1ab+ab ab +a^2b+ab^2 a^2b^2$, with $a^2=b^2=1$, $ab=ba$. PS: on the other hand, this is true if (and only if?) $G$ cyclic, since then you have an injective character $\chi : G\to \mathbb{C}^\times$, whose linear extension to $\mathbb{Z}[G]$ has nonzero value on you element. PPS: your expression is indeed $0$ in $\mathbb{Z}[G]$ whenever $G$ isn't cyclic. It is enough to shows it's zero in $\mathbb{C}[G]$, which (by elementary representation theory) is a product of fields $\mathbb{C}_\chi$ over the characters $\chi\in \hat{G}$. The result follows, since no character is injective when $G$ isn't cyclic. 

